12.4: Rotations

What if you wanted to find the center of rotation and angle of rotation for the arrows in the international recycling symbol below? It is three arrows rotated around a point. Let’s assume that the arrow on the top is the preimage and the other two are its images. Find the center of rotation and the angle of rotation for each image. After completing this Concept, you'll be able to answer these questions.

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Guidance

A
transformation
is an operation that moves, flips, or changes a figure to create a new figure. A
rigid transformation
is a transformation that preserves size and shape. The rigid transformations are: translations, reflections, and rotations (discussed here). The new figure created by a transformation is called the
image
. The original figure is called the
preimage
. Another word for a rigid transformation is an
isometry
. Rigid transformations are also called
congruence transformations
. If the preimage is
, then the image would be labeled
, said “a prime.” If there is an image of
, that would be labeled
, said “a double prime.”

A
rotation
is a transformation by which a figure is turned around a fixed point to create an image. The
center of rotation
is the fixed point that a figure is rotated around. Lines can be drawn from the preimage to the center of rotation, and from the center of rotation to the image. The angle formed by these lines is the
angle of rotation.

In this Concept, our center of rotation will always be the
origin.
Rotations can also be clockwise or counterclockwise. We will only do
counterclockwise
rotations, to go along with the way the quadrants are numbered.

Investigation: Drawing a Rotation of

Tools Needed: pencil, paper, protractor, ruler

Draw
and a point
outside the circle.

Draw the line segment
.

Take your protractor, place the center on
and the initial side on
. Mark a
angle.

Find
such that
.

Repeat steps 2-4 with points
and
.

Connect
and
to form
.

This is the process you would follow to rotate any figure
counterclockwise. If it was a different angle measure, then in Step 3, you would mark a different angle. You will need to repeat steps 2-4 for every vertex of the shape.

Common Rotations

Rotation of
:
If
is rotated
around the origin, then the image will be
.

Rotation of
:
If
is rotated
around the origin, then the image will be
.

Rotation of
:
If
is rotated
around the origin, then the image will be
.

While we can rotate any image any amount of degrees, only
and
have special rules. To rotate a figure by an angle measure other than these three, you must use the process from the Investigation.

Example A

Rotate
, with vertices
and
. Find the coordinates of
.

It is very helpful to graph the triangle. If
is
, that means it is 7 units to the right of the origin and 4 units up.
would then be 7 units to the
left
of the origin and 4 units
down.
The vertices are:

Example B

Rotate
counter-clockwise about the origin.

Using the
rotation rule,
is (8, 2).

Example C

Find the coordinates of
after a
rotation counter-clockwise about the origin.

Concept Problem Revisited

The center of rotation is shown in the picture below. If we draw rays to the same point in each arrow, we see that the two images are a
rotation in either direction.

Vocabulary

A
transformation
is an operation that moves, flips, or otherwise changes a figure to create a new figure. A
rigid transformation
(also known as an
isometry
or
congruence transformation
) is a transformation that does not change the size or shape of a figure. The new figure created by a transformation is called the
image
. The original figure is called the
preimage
. A
rotation
is a transformation where a figure is turned around a fixed point to create an image. The lines drawn from the preimage to the
center of rotation
and from the center of rotation to the image form the
angle of rotation
.

Guided Practice

1. The rotation of a quadrilateral is shown below. What is the measure of
and
?

2. A rotation of
clockwise is the same as what counterclockwise rotation?

3. A rotation of
counterclockwise is the same as what clockwise rotation?

Answers:

1. Because a rotation is an isometry that produces congruent figures, we can set up two equations to solve for
and
.

2. There are
around a point. So, an
rotation clockwise is the same as a
rotation counterclockwise.

3.
clockwise rotation.

Practice

In the questions below, every rotation is
counterclockwise,
unless otherwise stated.

If you rotated the letter
counterclockwise, what letter would you have?

If you rotated the letter
clockwise,
what letter would you have? Why do you think that is?

A
clockwise rotation is the same as what counterclockwise rotation?

A
clockwise rotation is the same as what counterclockwise rotation?

Rotating a figure
is the same as what other rotation?

Rotate each figure in the coordinate plane the given angle measure. The center of rotation is the origin.

Algebra Connection
Find the measure of
in the rotations below. The blue figure is the preimage.

Find the angle of rotation for the graphs below. The center of rotation is the origin and the blue figure is the preimage.